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Theorem elpcliN 35760
Description: Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
elpcli.s  |-  S  =  ( PSubSp `  K )
elpcli.c  |-  U  =  ( PCl `  K
)
Assertion
Ref Expression
elpcliN  |-  ( ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  /\  Q  e.  ( U `  X )
)  ->  Q  e.  Y )

Proof of Theorem elpcliN
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 simp1 996 . . . . . 6  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  K  e.  V )
2 simp2 997 . . . . . . 7  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  X  C_  Y )
3 eqid 2457 . . . . . . . . 9  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 elpcli.s . . . . . . . . 9  |-  S  =  ( PSubSp `  K )
53, 4psubssat 35621 . . . . . . . 8  |-  ( ( K  e.  V  /\  Y  e.  S )  ->  Y  C_  ( Atoms `  K ) )
653adant2 1015 . . . . . . 7  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  Y  C_  ( Atoms `  K )
)
72, 6sstrd 3509 . . . . . 6  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  X  C_  ( Atoms `  K )
)
8 elpcli.c . . . . . . 7  |-  U  =  ( PCl `  K
)
93, 4, 8pclvalN 35757 . . . . . 6  |-  ( ( K  e.  V  /\  X  C_  ( Atoms `  K
) )  ->  ( U `  X )  =  |^| { z  e.  S  |  X  C_  z } )
101, 7, 9syl2anc 661 . . . . 5  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( U `  X )  =  |^| { z  e.  S  |  X  C_  z } )
1110eleq2d 2527 . . . 4  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( Q  e.  ( U `  X )  <->  Q  e.  |^|
{ z  e.  S  |  X  C_  z } ) )
12 elintrabg 4301 . . . . 5  |-  ( Q  e.  |^| { z  e.  S  |  X  C_  z }  ->  ( Q  e.  |^| { z  e.  S  |  X  C_  z }  <->  A. z  e.  S  ( X  C_  z  ->  Q  e.  z )
) )
1312ibi 241 . . . 4  |-  ( Q  e.  |^| { z  e.  S  |  X  C_  z }  ->  A. z  e.  S  ( X  C_  z  ->  Q  e.  z ) )
1411, 13syl6bi 228 . . 3  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( Q  e.  ( U `  X )  ->  A. z  e.  S  ( X  C_  z  ->  Q  e.  z ) ) )
15 sseq2 3521 . . . . . . . 8  |-  ( z  =  Y  ->  ( X  C_  z  <->  X  C_  Y
) )
16 eleq2 2530 . . . . . . . 8  |-  ( z  =  Y  ->  ( Q  e.  z  <->  Q  e.  Y ) )
1715, 16imbi12d 320 . . . . . . 7  |-  ( z  =  Y  ->  (
( X  C_  z  ->  Q  e.  z )  <-> 
( X  C_  Y  ->  Q  e.  Y ) ) )
1817rspccv 3207 . . . . . 6  |-  ( A. z  e.  S  ( X  C_  z  ->  Q  e.  z )  ->  ( Y  e.  S  ->  ( X  C_  Y  ->  Q  e.  Y ) ) )
1918com13 80 . . . . 5  |-  ( X 
C_  Y  ->  ( Y  e.  S  ->  ( A. z  e.  S  ( X  C_  z  ->  Q  e.  z )  ->  Q  e.  Y ) ) )
2019imp 429 . . . 4  |-  ( ( X  C_  Y  /\  Y  e.  S )  ->  ( A. z  e.  S  ( X  C_  z  ->  Q  e.  z )  ->  Q  e.  Y ) )
21203adant1 1014 . . 3  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( A. z  e.  S  ( X  C_  z  ->  Q  e.  z )  ->  Q  e.  Y ) )
2214, 21syld 44 . 2  |-  ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  ->  ( Q  e.  ( U `  X )  ->  Q  e.  Y ) )
2322imp 429 1  |-  ( ( ( K  e.  V  /\  X  C_  Y  /\  Y  e.  S )  /\  Q  e.  ( U `  X )
)  ->  Q  e.  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811    C_ wss 3471   |^|cint 4288   ` cfv 5594   Atomscatm 35131   PSubSpcpsubsp 35363   PClcpclN 35754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-psubsp 35370  df-pclN 35755
This theorem is referenced by:  pclfinclN  35817
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